共查询到20条相似文献,搜索用时 15 毫秒
1.
We use the self-similar tilings constructed in (Pearse in Indiana Univ. Math J. 56(6):3151–3169, 2007) to define a generating function for the geometry of a self-similar set in Euclidean space. This tubularzeta function encodes
scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This
allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubularzeta function and hence
develop a tube formula for self-similar tilings in ℝd. The resulting power series in εis a fractal extension of Steiner’s classical tube formula for convex bodies K⊆ℝ
d
. Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i=0,1,…,d−1, just as Steiner’s does. However, our formula also contains a term for each complex dimension. This provides further justification
for the term “complex dimension”. It also extends several aspects of the theory of fractal strings to higher dimensions and
sheds new light on the tube formula for fractals strings obtained in (Lapidus and van Frankenhuijsen in Fractal Geometry,
Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2006). 相似文献
2.
We consider self-affine tilings in ℝ n with expansion matrix φ and address the question which matrices φ can arise this way. In one dimension, λ is an expansion factor of a self-affine tiling if and only if |λ| is a Perron number, by a result of Lind. In two dimensions, when φ is a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston: if a complex λ is an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition for φ to be an expansion matrix for any n, assuming only that φ is diagonalizable over ℂ. We conjecture that this condition on φ is also sufficient for the existence of a self-affine tiling. 相似文献
3.
Natalie M. Priebe 《Geometriae Dedicata》2000,79(3):239-265
In this paper, a technique for analyzing levels of hierarchy in a tiling
of Euclidean space is presented. Fixing a central configuration P of tiles in
, a `derived Voronoï' tessellation
P is constructed based on the locations of copies of P in
. A family of derived Voronoï tilings
is formed by allowing the central configurations to vary through an infinite number of possibilities. The family
will normally be an infinite one, but we show that for a self-similar tiling
it is finite up to similarity. In addition, we show that if the family
is finite up to similarity, then
is pseudo-self-similar. The relationship between self-similarity and pseudo-self-similarity is not well understood, and this is the obstruction to a complete characterization of self-similarity via our method. A discussion and conjecture on the connection between the two forms of hierarchy for tilings is provided. 相似文献
4.
谱与Tilings分别在分析与几何中起着重要作用,它们之间的关系困惑了数学家好长时间.许多猜想,例如Fuglede谱集猜测等,都再次强调建立两者之间联系的重要性,尽管所期待的结果还没有得到.在这篇文章里,我们得到谱与Tilings的一些特征性质,目的在于阐明两者之间的某些共轭关系. 相似文献
5.
Carl Ludwig Siegel 《Mathematische Annalen》1937,114(1):57-68
Ohne ZusammenfassungGeschrieben in Dankbarkeit und Verehrung für Edmund Landau zu seinem 60. Geburtstag am 14. Februar 1937 相似文献
6.
We consider the ? n -Galois covering ?? n of the algebra A introduced by F. Xu [Adv. Math., 2008, 219: 1872?C1893]. We calculate the dimensions of all Hochschild cohomology groups of ?? n and give the ring structure of the Hochschild cohomology ring modulo nilpotence. As a conclusion, we provide a class of counterexamples to Snashall-Solberg??s conjecture. 相似文献
7.
Ivan Stanimirovi? 《Journal of Applied Mathematics and Computing》2012,40(1-2):569-586
Full-rank block LDL ? decomposition of a Hermitian n×n block matrix A is examined, where the iterative procedure evaluating the sub-matrices appearing in L and D is provided. This factorization is used to evaluate the inverse and Moore-Penrose inverse of a Hermitian n×n block matrix. The method for the calculation of the Moore-Penrose inverse of an arbitrary 2×2 block matrix is also provided. Therefore, matrix products A ? A and AA ? and the corresponding full-rank block LDL ? factorizations are observed. Also, a simple explicit formulae calculating the solution vector components of the normal system of equations is stated, where the LDL ? decomposition of the system matrix is done. 相似文献
8.
It is known that the orbit closures for the representations of the equioriented Dynkin quivers ? n are normal and Cohen–Macaulay varieties with rational singularities. In the paper we prove the same for the Dynkin quivers ? n with arbitrary orientation. Received: 25 October 2000 / Revised version: 28 February 2001 相似文献
9.
10.
Alexander Koldobsky 《Discrete and Computational Geometry》2012,47(3):538-547
Let 2≤n≤4. We show that for an arbitrary measure μ with even continuous density in ℝ
n
and any origin-symmetric convex body K in ℝ
n
,
m(K) £ \fracnn-1\frac|B2n|\fracn-1n|B2n-1|maxx ? Sn-1 m(K?x^)\operatornameVoln(K)1/n,\mu(K) \le\frac{n}{n-1}\frac{|B_2^n|^{\frac{n-1}{n}}}{|B_2^{n-1}|}\max_{\xi\in S^{n-1}} \mu\bigl(K\cap\xi^\bot\bigr)\operatorname{Vol}_n(K)^{1/n}, 相似文献
11.
B. F. Skubenko 《Journal of Mathematical Sciences》1992,62(4):2928-2935
The following theorem is proved: if for all
(X0)one has ¦ F(x) ¦ >0, where F(x) is a decomposable form of degree n of n variables, then, for n 3, F(x) is proportional to an integral form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 183, pp. 142–154, 1990. 相似文献
12.
A method for evaluating Hilbert transforms, by means of Turán quadrature rules with generalized Gegenbauer weights, is presented. The main feature of these integration formulas is the independence of the nodes of their multiplicity and thus of the precision degree. The error is analyzed both from a real and a complex perspective; in this context a new representation of the remainder term of the quadrature rules with multiple nodes for the evaluation of Hilbert transforms, valid not only for the particular class of weight functions here considered, is presented. A few numerical examples are provided.Work sponsored by Ministero della Ricerca Scientifica e Tecnologia, Italy. 相似文献
13.
William C. Brown 《代数通讯》2013,41(10):4051-4066
14.
C. G. Venketasubramanian 《Israel Journal of Mathematics》2013,194(1):1-44
For a nonarchimedean local field F, let GL(n):= GL(n, F) and GL(n?1) be embedded in GL(n) via g ? ( 0 1 g 0 ). Let π be an irreducible admissible representation of GL(n) for n ≥ 3. We prove that π is GL(n ? 1)-distinguished if and only if the Langlands parameter L(π) associated to π by the Local Langlands Correspondence has a subrepresentation L(11 n?2) of dimension n?2 corresponding to the trivial representation of GL(n?2) such that the two-dimensional quotient L(π)/L(11 n?2) corresponds either to an infinite-dimensional representation or the one-dimensional representations $\nu ^{ \pm (\tfrac{{n - 2}}{2})} $ of GL(2). We also prove that, for a parabolic subgroup P of GL(n) and an irreducible admissible representation ρ of the Levi subgroup of P, $\dim _\mathbb{C} (Hom_{GL(n - 1)} [ind_P^{GL(n)} (\rho ),\mathbb{I}_{n - 1} ]) \leqslant 2$ . For the standard Borel subgroup B n of GL(n) and characters x i of GL(1), we classify all representations ξ of the form $ind_{B_n }^{GL(n)} (\chi _1 \otimes \cdots \otimes \chi _n )$ for which $\dim _\mathbb{C} (Hom_{GL(n - 1)} [\xi ,\mathbb{I}_{n - 1} ]) = 2$ . 相似文献
15.
Nicolas Eisen 《Arkiv f?r Matematik》2012,50(1):69-87
We give a holomorphic extension result for continuous CR functions on a non-generic CR submanifold N of ℂ
n
to complex transversal wedges with edges containing N. We show that given any v∈ℂ
n
∖(T
p
N+iT
p
N), there exists a wedge of direction v whose edge contains a neighborhood of p in N, such that any continuous CR function defined locally near p extends holomorphically to that wedge. 相似文献
16.
17.
18.
A group representation method is used to construct minimal isometric embeddings 2
2 into 8
10 and 10
12 over C. The second of them yields a tight 5-design in C
2. The corresponding angle set contains some irrational numbers. 相似文献
19.
20.
A t-cover of a quadric
is a set C of t-dimensional subspaces contained in
such that every point of
is contained in at least one element of C.We consider (n – 1)-covers of the hyperbolic quadric Q
+(2n + 1, q). We show that such a cover must have at least q
n + 1 + 2q + 1 elements, give an example of this size for even q and describe what covers of this size should look like. 相似文献
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